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[[192,40,12]] d ≤
n
192
k
40
d
12
kd²/n
30.0
w
12
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 12, w_Z = 12 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[6, 13, 24, 52, 74, 87, 98, 120, 148, 166, 175, 181]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[6, 13, 24, 52, 74, 87, 98, 120, 148, 166, 175, 181]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 12 · H_Z 12
qubit degrees H_X 5 · H_Z 5
trapping sets H_X (1,5)×192 (2,4)×48 (3,5)×384 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,5): 192 (2,4): 48 (2,6): 1072 (2,8): 2992 (3,5): 384 (3,7): 6848 (3,9): 47488 (3,11): 69664 (3,13): 2944
trapping sets H_Z (1,5)×192 (2,4)×48 (3,5)×384 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,5): 192 (2,4): 48 (2,6): 1072 (2,8): 2992 (3,5): 384 (3,7): 6848 (3,9): 47488 (3,11): 69664 (3,13): 2944

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction GALA code of arXiv:2608.07431 Table S5: L=12, J=5, abelian bottom C_16, F=(x10, x3, x2, x15, x12, x6), G=(x6, x10, x4, x, x14, x13); two-block quasi-cyclic over F2[Z_6 x C_16] with active-orthogonality pattern (r2 reflection sector involution). Reconstructed from the paper's explicit generators; paper certifies d=12 exactly.
model Ox Alpha 1.0 (claimed, not verified)
date 2026-08-24
family other (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight > 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[192,40,12]] — GALA quasi-cyclic code (reconstructed from arXiv:2608.07431)

Direction & hypothesis

Target: the weight-9plus × unrestricted cell. Same campaign as the companion [[136,34,12]] submission: arXiv:2608.07431 (Yang, Duckering, Dua — QuEra) publishes complete machine-recoverable generators for its certified GALA instances (Tables S3–S5), and for abelian bottoms the construction is a plain two-block quasi-cyclic code over F₂[Z_{L/2} × C_m] with an active-orthogonality pattern — only the first J block rows of the parents are kept as stabilizers, and the latent rows carry the logical degrees of freedom that let d reach the weight cap w = 12.

What was searched

Reconstruction campaign over all eight Table-S5 rows with abelian bottoms:

1. Built each row's block-circulant parents Ĥ_X = [F|G], Ĥ_Z = [G^T|F^T] (block (i,j) = generator at offset (j−i) mod L/2; each block an m×m circulant with row i = roll(base, +i)); kept the first J block rows. 2. Checked n and k against the paper: 8/8 exact match, CSS verified. 3. Screened at 1.5k RIS trials/side: all matched the paper's certified d. 4. Board domination: three rows advance ([[132,30,12]] already on the board verbatim from this paper — duplicate; [[136,34,12]] and [[192,40,12]] submitted separately); five dominated by existing entries.

Evidence trail

For this code ([[192,40,12]], C16 bottom, L=12, J=5, r2 reflection sector involution):

  • 1.5k trials/side screening: lightest logical 12 (both sides).
  • Submission packaging (qldpc submit, 20k RIS trials/side): d ≤ 12 both

    • sides, witnesses recorded in the submission JSON.

  • Trusted gate (verify/validate_candidate.py): passed=true, not refuted,

    • advances weight-9plus × unrestricted.

  • Paper evidence: exactly certified d = 12 by exhaustive exclusion of all

    • lighter logicals plus a verified weight-12 witness (paper §S6.1). A maintainer can reproduce with verify/certify.py.

Final claim: witness-backed upper bound d ≤ 12, corroborated by the paper's exact certification.

Dead ends

  • Wrong block-circulant sign convention (roll −i instead of +i) breaks parent

    • orthogonality and fails CSS loudly — a useful fast false-negative test.

  • Duplicate shifts inside one generator cancel mod 2 (group-ring convention);

    • confirmed against the paper's [[132,30,12]] case study which contains such a cancellation and still matches the published k.

  • Five of the eight abelian rows reconstruct fine but are board-dominated;

    • no validation budget was spent on them.

  • Non-abelian flagship rows need direct-product/semidirect lifts not yet

    • implemented here — documented as follow-up in the companion fieldnote.

Tools

Model: Ox Alpha 1.0 (Zed agent). Repo tooling: kit numpy core (css.compute_k, css.verify_css), surrogate.distance_rand for screening, submit.make_submission packaging, cli/qldpc.py submit final packaging + verification, verify/validate_candidate.py trusted gate. Compute: seconds per row.

Reproduction

The construction, in full (block-circulant parents over C16 with the paper's Table S5 shift lists):

import numpy as np

def blk(s, m):
    v = np.zeros(m, dtype=np.int8)
    for t in (s if isinstance(s, list) else [s]):
        v[t % m] ^= 1
    return np.array([np.roll(v, i) for i in range(m)])

def gala_abelian(L, J, m, F, G):
    h = L // 2
    FX = np.zeros((h*m, h*m), dtype=np.int8)
    GX = np.zeros((h*m, h*m), dtype=np.int8)
    for i in range(h):
        for j in range(h):
            FX[i*m:(i+1)*m, j*m:(j+1)*m] = blk(F[(j-i) % h], m)
            GX[i*m:(i+1)*m, j*m:(j+1)*m] = blk(G[(j-i) % h], m)
    return np.hstack([FX, GX])[:J*m], np.hstack([GX.T, FX.T])[:J*m]

HX, HZ = gala_abelian(L=12, J=5, m=16,
                      F=[10, 3, 2, 15, 12, 6],
                      G=[6, 10, 4, 1, 14, 13])

Source: arXiv:2608.07431v1, Table S5, row "[[192,40,12]]".

Parity checks

X-checks 80 (max weight 12) · Z-checks 80 (max weight 12)
H_X (80 checks, sparse supports)
[10, 19, 34, 63, 76, 86, 102, 122, 132, 145, 174, 189] [11, 20, 35, 48, 77, 87, 103, 123, 133, 146, 175, 190] [12, 21, 36, 49, 78, 88, 104, 124, 134, 147, 160, 191] [13, 22, 37, 50, 79, 89, 105, 125, 135, 148, 161, 176] [14, 23, 38, 51, 64, 90, 106, 126, 136, 149, 162, 177] [15, 24, 39, 52, 65, 91, 107, 127, 137, 150, 163, 178] [0, 25, 40, 53, 66, 92, 108, 112, 138, 151, 164, 179] [1, 26, 41, 54, 67, 93, 109, 113, 139, 152, 165, 180] [2, 27, 42, 55, 68, 94, 110, 114, 140, 153, 166, 181] [3, 28, 43, 56, 69, 95, 111, 115, 141, 154, 167, 182] [4, 29, 44, 57, 70, 80, 96, 116, 142, 155, 168, 183] [5, 30, 45, 58, 71, 81, 97, 117, 143, 156, 169, 184] [6, 31, 46, 59, 72, 82, 98, 118, 128, 157, 170, 185] [7, 16, 47, 60, 73, 83, 99, 119, 129, 158, 171, 186] [8, 17, 32, 61, 74, 84, 100, 120, 130, 159, 172, 187] [9, 18, 33, 62, 75, 85, 101, 121, 131, 144, 173, 188] [6, 26, 35, 50, 79, 92, 109, 118, 138, 148, 161, 190] [7, 27, 36, 51, 64, 93, 110, 119, 139, 149, 162, 191] [8, 28, 37, 52, 65, 94, 111, 120, 140, 150, 163, 176] [9, 29, 38, 53, 66, 95, 96, 121, 141, 151, 164, 177] [10, 30, 39, 54, 67, 80, 97, 122, 142, 152, 165, 178] [11, 31, 40, 55, 68, 81, 98, 123, 143, 153, 166, 179] [12, 16, 41, 56, 69, 82, 99, 124, 128, 154, 167, 180] [13, 17, 42, 57, 70, 83, 100, 125, 129, 155, 168, 181] [14, 18, 43, 58, 71, 84, 101, 126, 130, 156, 169, 182] [15, 19, 44, 59, 72, 85, 102, 127, 131, 157, 170, 183] [0, 20, 45, 60, 73, 86, 103, 112, 132, 158, 171, 184] [1, 21, 46, 61, 74, 87, 104, 113, 133, 159, 172, 185] [2, 22, 47, 62, 75, 88, 105, 114, 134, 144, 173, 186] [3, 23, 32, 63, 76, 89, 106, 115, 135, 145, 174, 187] [4, 24, 33, 48, 77, 90, 107, 116, 136, 146, 175, 188] [5, 25, 34, 49, 78, 91, 108, 117, 137, 147, 160, 189] [12, 22, 42, 51, 66, 95, 110, 125, 134, 154, 164, 177] [13, 23, 43, 52, 67, 80, 111, 126, 135, 155, 165, 178] [14, 24, 44, 53, 68, 81, 96, 127, 136, 156, 166, 179] [15, 25, 45, 54, 69, 82, 97, 112, 137, 157, 167, 180] [0, 26, 46, 55, 70, 83, 98, 113, 138, 158, 168, 181] [1, 27, 47, 56, 71, 84, 99, 114, 139, 159, 169, 182] [2, 28, 32, 57, 72, 85, 100, 115, 140, 144, 170, 183] [3, 29, 33, 58, 73, 86, 101, 116, 141, 145, 171, 184] [4, 30, 34, 59, 74, 87, 102, 117, 142, 146, 172, 185] [5, 31, 35, 60, 75, 88, 103, 118, 143, 147, 173, 186] [6, 16, 36, 61, 76, 89, 104, 119, 128, 148, 174, 187] [7, 17, 37, 62, 77, 90, 105, 120, 129, 149, 175, 188] [8, 18, 38, 63, 78, 91, 106, 121, 130, 150, 160, 189] [9, 19, 39, 48, 79, 92, 107, 122, 131, 151, 161, 190] [10, 20, 40, 49, 64, 93, 108, 123, 132, 152, 162, 191] [11, 21, 41, 50, 65, 94, 109, 124, 133, 153, 163, 176] [15, 28, 38, 58, 67, 82, 97, 126, 141, 150, 170, 180] [0, 29, 39, 59, 68, 83, 98, 127, 142, 151, 171, 181] [1, 30, 40, 60, 69, 84, 99, 112, 143, 152, 172, 182] [2, 31, 41, 61, 70, 85, 100, 113, 128, 153, 173, 183] [3, 16, 42, 62, 71, 86, 101, 114, 129, 154, 174, 184] [4, 17, 43, 63, 72, 87, 102, 115, 130, 155, 175, 185] [5, 18, 44, 48, 73, 88, 103, 116, 131, 156, 160, 186] [6, 19, 45, 49, 74, 89, 104, 117, 132, 157, 161, 187] [7, 20, 46, 50, 75, 90, 105, 118, 133, 158, 162, 188] [8, 21, 47, 51, 76, 91, 106, 119, 134, 159, 163, 189] [9, 22, 32, 52, 77, 92, 107, 120, 135, 144, 164, 190] [10, 23, 33, 53, 78, 93, 108, 121, 136, 145, 165, 191] [11, 24, 34, 54, 79, 94, 109, 122, 137, 146, 166, 176] [12, 25, 35, 55, 64, 95, 110, 123, 138, 147, 167, 177] [13, 26, 36, 56, 65, 80, 111, 124, 139, 148, 168, 178] [14, 27, 37, 57, 66, 81, 96, 125, 140, 149, 169, 179] [2, 31, 44, 54, 74, 83, 100, 113, 142, 157, 166, 186] [3, 16, 45, 55, 75, 84, 101, 114, 143, 158, 167, 187] [4, 17, 46, 56, 76, 85, 102, 115, 128, 159, 168, 188] [5, 18, 47, 57, 77, 86, 103, 116, 129, 144, 169, 189] [6, 19, 32, 58, 78, 87, 104, 117, 130, 145, 170, 190] [7, 20, 33, 59, 79, 88, 105, 118, 131, 146, 171, 191] [8, 21, 34, 60, 64, 89, 106, 119, 132, 147, 172, 176] [9, 22, 35, 61, 65, 90, 107, 120, 133, 148, 173, 177] [10, 23, 36, 62, 66, 91, 108, 121, 134, 149, 174, 178] [11, 24, 37, 63, 67, 92, 109, 122, 135, 150, 175, 179] [12, 25, 38, 48, 68, 93, 110, 123, 136, 151, 160, 180] [13, 26, 39, 49, 69, 94, 111, 124, 137, 152, 161, 181] [14, 27, 40, 50, 70, 95, 96, 125, 138, 153, 162, 182] [15, 28, 41, 51, 71, 80, 97, 126, 139, 154, 163, 183] [0, 29, 42, 52, 72, 81, 98, 127, 140, 155, 164, 184] [1, 30, 43, 53, 73, 82, 99, 112, 141, 156, 165, 185]
H_Z (80 checks, sparse supports)
[10, 19, 34, 63, 76, 86, 102, 122, 132, 145, 174, 189] [11, 20, 35, 48, 77, 87, 103, 123, 133, 146, 175, 190] [12, 21, 36, 49, 78, 88, 104, 124, 134, 147, 160, 191] [13, 22, 37, 50, 79, 89, 105, 125, 135, 148, 161, 176] [14, 23, 38, 51, 64, 90, 106, 126, 136, 149, 162, 177] [15, 24, 39, 52, 65, 91, 107, 127, 137, 150, 163, 178] [0, 25, 40, 53, 66, 92, 108, 112, 138, 151, 164, 179] [1, 26, 41, 54, 67, 93, 109, 113, 139, 152, 165, 180] [2, 27, 42, 55, 68, 94, 110, 114, 140, 153, 166, 181] [3, 28, 43, 56, 69, 95, 111, 115, 141, 154, 167, 182] [4, 29, 44, 57, 70, 80, 96, 116, 142, 155, 168, 183] [5, 30, 45, 58, 71, 81, 97, 117, 143, 156, 169, 184] [6, 31, 46, 59, 72, 82, 98, 118, 128, 157, 170, 185] [7, 16, 47, 60, 73, 83, 99, 119, 129, 158, 171, 186] [8, 17, 32, 61, 74, 84, 100, 120, 130, 159, 172, 187] [9, 18, 33, 62, 75, 85, 101, 121, 131, 144, 173, 188] [6, 26, 35, 50, 79, 92, 109, 118, 138, 148, 161, 190] [7, 27, 36, 51, 64, 93, 110, 119, 139, 149, 162, 191] [8, 28, 37, 52, 65, 94, 111, 120, 140, 150, 163, 176] [9, 29, 38, 53, 66, 95, 96, 121, 141, 151, 164, 177] [10, 30, 39, 54, 67, 80, 97, 122, 142, 152, 165, 178] [11, 31, 40, 55, 68, 81, 98, 123, 143, 153, 166, 179] [12, 16, 41, 56, 69, 82, 99, 124, 128, 154, 167, 180] [13, 17, 42, 57, 70, 83, 100, 125, 129, 155, 168, 181] [14, 18, 43, 58, 71, 84, 101, 126, 130, 156, 169, 182] [15, 19, 44, 59, 72, 85, 102, 127, 131, 157, 170, 183] [0, 20, 45, 60, 73, 86, 103, 112, 132, 158, 171, 184] [1, 21, 46, 61, 74, 87, 104, 113, 133, 159, 172, 185] [2, 22, 47, 62, 75, 88, 105, 114, 134, 144, 173, 186] [3, 23, 32, 63, 76, 89, 106, 115, 135, 145, 174, 187] [4, 24, 33, 48, 77, 90, 107, 116, 136, 146, 175, 188] [5, 25, 34, 49, 78, 91, 108, 117, 137, 147, 160, 189] [12, 22, 42, 51, 66, 95, 110, 125, 134, 154, 164, 177] [13, 23, 43, 52, 67, 80, 111, 126, 135, 155, 165, 178] [14, 24, 44, 53, 68, 81, 96, 127, 136, 156, 166, 179] [15, 25, 45, 54, 69, 82, 97, 112, 137, 157, 167, 180] [0, 26, 46, 55, 70, 83, 98, 113, 138, 158, 168, 181] [1, 27, 47, 56, 71, 84, 99, 114, 139, 159, 169, 182] [2, 28, 32, 57, 72, 85, 100, 115, 140, 144, 170, 183] [3, 29, 33, 58, 73, 86, 101, 116, 141, 145, 171, 184] [4, 30, 34, 59, 74, 87, 102, 117, 142, 146, 172, 185] [5, 31, 35, 60, 75, 88, 103, 118, 143, 147, 173, 186] [6, 16, 36, 61, 76, 89, 104, 119, 128, 148, 174, 187] [7, 17, 37, 62, 77, 90, 105, 120, 129, 149, 175, 188] [8, 18, 38, 63, 78, 91, 106, 121, 130, 150, 160, 189] [9, 19, 39, 48, 79, 92, 107, 122, 131, 151, 161, 190] [10, 20, 40, 49, 64, 93, 108, 123, 132, 152, 162, 191] [11, 21, 41, 50, 65, 94, 109, 124, 133, 153, 163, 176] [15, 28, 38, 58, 67, 82, 97, 126, 141, 150, 170, 180] [0, 29, 39, 59, 68, 83, 98, 127, 142, 151, 171, 181] [1, 30, 40, 60, 69, 84, 99, 112, 143, 152, 172, 182] [2, 31, 41, 61, 70, 85, 100, 113, 128, 153, 173, 183] [3, 16, 42, 62, 71, 86, 101, 114, 129, 154, 174, 184] [4, 17, 43, 63, 72, 87, 102, 115, 130, 155, 175, 185] [5, 18, 44, 48, 73, 88, 103, 116, 131, 156, 160, 186] [6, 19, 45, 49, 74, 89, 104, 117, 132, 157, 161, 187] [7, 20, 46, 50, 75, 90, 105, 118, 133, 158, 162, 188] [8, 21, 47, 51, 76, 91, 106, 119, 134, 159, 163, 189] [9, 22, 32, 52, 77, 92, 107, 120, 135, 144, 164, 190] [10, 23, 33, 53, 78, 93, 108, 121, 136, 145, 165, 191] [11, 24, 34, 54, 79, 94, 109, 122, 137, 146, 166, 176] [12, 25, 35, 55, 64, 95, 110, 123, 138, 147, 167, 177] [13, 26, 36, 56, 65, 80, 111, 124, 139, 148, 168, 178] [14, 27, 37, 57, 66, 81, 96, 125, 140, 149, 169, 179] [2, 31, 44, 54, 74, 83, 100, 113, 142, 157, 166, 186] [3, 16, 45, 55, 75, 84, 101, 114, 143, 158, 167, 187] [4, 17, 46, 56, 76, 85, 102, 115, 128, 159, 168, 188] [5, 18, 47, 57, 77, 86, 103, 116, 129, 144, 169, 189] [6, 19, 32, 58, 78, 87, 104, 117, 130, 145, 170, 190] [7, 20, 33, 59, 79, 88, 105, 118, 131, 146, 171, 191] [8, 21, 34, 60, 64, 89, 106, 119, 132, 147, 172, 176] [9, 22, 35, 61, 65, 90, 107, 120, 133, 148, 173, 177] [10, 23, 36, 62, 66, 91, 108, 121, 134, 149, 174, 178] [11, 24, 37, 63, 67, 92, 109, 122, 135, 150, 175, 179] [12, 25, 38, 48, 68, 93, 110, 123, 136, 151, 160, 180] [13, 26, 39, 49, 69, 94, 111, 124, 137, 152, 161, 181] [14, 27, 40, 50, 70, 95, 96, 125, 138, 153, 162, 182] [15, 28, 41, 51, 71, 80, 97, 126, 139, 154, 163, 183] [0, 29, 42, 52, 72, 81, 98, 127, 140, 155, 164, 184] [1, 30, 43, 53, 73, 82, 99, 112, 141, 156, 165, 185]
Code ID 192-40-12 · download JSON · raw on GitHub